The volumetric flow rate \(Q\)  shows a number of scaling symmetries. We now combine these with the Euler theorem for homogeneous functions
to deduce the functional form of \(Q=Q(\Delta P,\alpha ,\beta ,\lbrace q_c\rbrace ,N_x,N_y)\)  [1]}. Here \(\lbrace q_c\rbrace \)  is the set of thresholds, one for each link in the network. The volumetric flow rate is extensive in the width of the network, \(N_y\) . Hence,
\(&&Q(\Delta P,\alpha ,\beta ,\lbrace q_c\rbrace ,N_x,\lambda _y N_y)\nonumber \\&=&\lambda _y Q(\Delta P,\alpha ,\beta ,\lbrace q_c\rbrace ,N_x,N_y)\;.\) 
